nLab spectrification of a sequential spectrum type



Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
logical conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type/path type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
completely presented setdiscrete object/0-truncated objecth-level 2-type/set/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels



Stable Homotopy theory



In classical algebraic topology, there is a spectrification functor which is left adjoint to the inclusion of spectra in prespectra. For instance, this is how a suspension spectrum is constructed: by spectrifying the prespectrum X nΣ nAX_n \coloneqq \Sigma^n A. There is an analogue in homotopy type theory which constructs the spectrification of a sequential spectrum type (i.e. prespectrum types) into a Omega-spectrum type (i.e. spectrum types).


The following higher inductive type should construct spectrification in homotopy type theory (though this has not yet been verified formally). (There are some abuses of notation below, which can be made precise using Coq typeclasses and implicit arguments.)

Inductive spectrify (X : prespectrum) : nat -> Type :=
| to_spectrify : forall n, X n -> spectrify X n
| spectrify_glue : forall n, spectrify X n ->
    to_spectrify (S n) (pt (S n)) == to_spectrify (S n) (pt (S n))
| to_spectrify_is_prespectrum_map : forall n (x : X n),
    spectrify_glue n (to_spectrify n x)
    == loop_functor (to_spectrify (S n)) (glue n x)
| spectrify_glue_retraction : forall n
    (p : to_spectrify (S n) (pt (S n)) == to_spectrify (S n) (pt (S n))),
    spectrify X n
| spectrify_glue_retraction_is_retraction : forall n (sx : spectrify X n),
    spectrify_glue_retraction n (spectrify_glue n sx) == sx
| spectrify_glue_section : forall n
    (p : to_spectrify (S n) (pt (S n)) == to_spectrify (S n) (pt (S n))),
    spectrify X n
| spectrify_glue_section_is_section : forall n
    (p : to_spectrify (S n) (pt (S n)) == to_spectrify (S n) (pt (S n))),
    spectrify_glue n (spectrify_glue_section n p) == p.

We can unravel this as follows, using more traditional notation. Let LXL X denote the spectrification being constructed. The first constructor says that each (LX) n(L X)_n comes with a map from X nX_n, called n\ell_n say (denoted to_spectrify n above). This induces a basepoint in each type (LX) n(L X)_n, namely the image n(*)\ell_n(*) of the basepoint of X nX_n. The many occurrences of

to_spectrify (S n) (pt (S n)) == to_spectrify (S n) (pt (S n))

simply refer to the based loop space of Ω n+1(*)(LX) n+1\Omega_{\ell_{n+1}(*)} (L X)_{n+1} of (LX) n+1(L X)_{n+1} at this base point.

Thus, the second constructor spectrify_glue gives the structure maps (LX) nΩ(LX) n+1(L X)_n \to \Omega (L X)_{n+1} to make LXL X into a prespectrum. Similarly, the third constructor says that the maps n:X n(LX) n\ell_n\colon X_n \to (L X)_n commute with the structure maps up to a specified homotopy.

Since the basepoints of the types (LX) n(L X)_n are induced from those of each X nX_n, this automatically implies that the maps (LX) nΩ(LX) n+1(L X)_n \to \Omega (L X)_{n+1} are pointed maps (up to a specified homotopy) and that the n\ell_n commute with these pointings (up to a specified homotopy). This makes \ell into a map of prespectra.

Finally, the fourth through seventh constructors say that LXL X is a spectrum, by giving h-isomorphism data: a retraction and a section for each glue map (LX) nΩ(LX) n+1(L X)_n \to \Omega (L X)_{n+1}. We could use adjoint equivalence data as we did for localization, but this approach avoids the presence of level-3 path constructors. (We could have used h-iso data in localization too, thereby avoiding even level-2 constructors there.) It is important, in general, to use a sort of equivalence data which forms an h-prop; otherwise we would be adding structure rather than merely the property of such-and-such map being an equivalence.

See also


Last revised on June 9, 2022 at 01:58:13. See the history of this page for a list of all contributions to it.